Exercise sheet 3 Patrick Loiseau, Paul de Kerret Game Theory, Fall - PDF document

Exercise sheet 3 Patrick Loiseau, Paul de Kerret Game Theory, Fall 2016 Exercise 1: Bertrand duopoly In class, we considered Cournot competition where two firms choose quantities and let the price be fixed by the market. We consider here a different model of competition on prices called Bertrand competition. Consider 2 firms producing identical products (i.e., that are perfect substitutes) with a constant marginal cost c . Each firm i choose the price p i ∈ [0 , 1] . We assume that the demand curve is linear so that the total demand is Q ( p ) = 1 − p where p = min( p 1 , p 2 ) and is entirely directed to the firm with the smallest price. If both prices are equal, the demand is equally shared between the two firms. We denote by q i the demand to firm i . For instance, the demand to firm 1 is  1 − p 1 if p 1 < p 2 ,  q 1 ( p 1 , p 2 ) = 0 if p 1 > p 2 , (1 − p 1 ) / 2 if p 1 = p 2 .  Note that this corresponds to the demand model used in class for Cournot duopoly with a = b = 1 . Both firms maximize their profit: u 1 ( p 1 , p 2 ) = p 1 q 1 − cq 1 , u 2 ( p 1 , p 2 ) = p 2 q 2 − cq 2 . 1. Suppose first that firm 1 is alone in the market. What is the optimal price p 1 ? What is the quantity purchased q 1 ? Comment. Answer: p 1 = (1 + c ) / 2 , q 1 = (1 − c ) / 2 . This is the monopolist price/quantity, the same seen in class in the Cournot model (normal since the models are the same if there is only one firm). 2. Compute the best response of firm 1 to a price p 2 of firm 2. [hint: no calculation, just reasoning.] Answer: BR 1 ( p 2 < c ) = ( p 2 , 1] (exit the market). BR 1 ( p 2 = c ) = [ p 2 , 1] . BR 1 ( p 2 ∈ ( c, p mono ) = p 2 − ǫ . BR 1 ( p 2 > p mono ) = p mono . 3. Find the Nash equilibria. Comment. Answer: Only one Nash equilibrium: ( c, c ) , i.e., price at marginal cost like in perfect competition. Comments: The result of question 3. is not very realistic. It is due to the modeling assumption that products are perfect substitutes. In the next exercise, we remove this assumption by using the linear city model. 1

Exercise 2: The linear city model In this exercise, we consider a duopoly where firms compete on prices (like in Bertrand duopoly) but we relax the assumption that products are identical. We think of a city as a line of length 1. There are 2 firms, at each end of the line. Each firm produces a product at constant marginal cost c . The firms simultaneously choose the price of their product p 1 and p 2 . Potential customers are uniformly distributed along the line, with a total population normalized to 1. Each potential customer buys exactly one unit either from firm 1 or firm 2, so the total demand is 1. Given its position on the line y (i.e., the customer is at distance y from firm 1 and 1 − y from firm 2), a customer will buy its unit: from firm 1 if p 1 + ty 2 < p 2 + t (1 − y ) 2 from firm 2 if p 1 + ty 2 > p 2 + t (1 − y ) 2 (1) and will toss a fair coin is these two quantities are equal. Conditions (1) translate what the customers care about: they try to buy from the firm with smallest price + t ( distance ) 2 . If we think of the line as representing geographical distance, t ( distance ) 2 is a transport cost. However, other interpretations are possible. For instance, we can think of the line as representing some aspect of the product quality. Then, the proximity to a firm represents the preference and the term t ( distance ) 2 measures the inconvenience of having to move away from the preferred product. Parameter t represents how much the products are differentiated . If t = 0 , the products are identical. The demand resulting from customers choices to firm i is denoted d i ( p 1 , p 2 ) . As usual, each firm’s objective is to maximize its profit: u 1 ( p 1 , p 2 ) = p 1 d 1 ( p 1 , p 2 ) − cd 1 ( p 1 , p 2 ) , u 2 ( p 1 , p 2 ) = p 2 d 2 ( p 1 , p 2 ) − cd 2 ( p 1 , p 2 ) . 1. Will either firm i ever set a price p i < c ? Why? Answer: No. Gets negative profit. Any price at least c is better. 2. Suppose that firm 2 sets a price p 2 . What is the maximum price p 1 for which firm 1 captures the entire market (i.e., has demand 1)? Answer: p 2 − t . 3. Suppose that prices p 1 and p 2 are close enough that each firm has positive demand. Find the position of the customer who is indifferent between buying from firm 1 and buying from firm 2. Compute the demand to firm 1 d 1 ( p 1 , p 2 ) . Answer: y = ( p 2 + t − p 1 ) / (2 t ) = d 1 ( p 1 , p 2 ) . 4. Suppose that prices p 1 and p 2 are close enough that each firm has positive demand. Compute the best-response of firm 1 to p 2 . Answer: BR 1 ( p 2 ) = ( p 2 + t + c ) / 2 . 2

5. Draw the complete best response diagram (be careful while plotting BR 1 ( p 2 ) when p 2 < c − t and p 2 > 3 t + c and similarly for BR 2 ). 6. Find the Nash equilibrium. Answer: ( t + c, t + c ) . 7. What is the equilibrium when t = 0 ? Comment. Answer: ( c, c ) . Same as Bertrand. Comments: Products differentiation allows firms to charge higher prices and make more profit. 3